This following is a small part of a rather large problem that has been bugging me.
Let $\Omega$ be any bounded open set in $\mathbb{R}^{2}$ of finite perimeter. Is it necessarily true that
$$\mathcal{H}^{1}(\partial (\Omega + B_{\delta})) \leq \mathcal{H}^{1}(\partial \Omega)+\mathcal{H}^{1}(\partial B_{\delta}) = \mathcal{H}^{1}(\partial \Omega)+2\pi\delta?$$
Here $\mathcal{H}^{1}$ is the Hausdorff measure, $B_{\delta}$ is the ball of radius $\delta$ and $ \Omega + B_{\delta}$ is the Minkowski sum of $\Omega$ and $B_{\delta}$. Here $\partial \Omega$ denotes the topological boundary of $\Omega$.
The question is motivated by the a note somewhere that we have equality if $\Omega$ is convex. I can't seem to construct a counter example but nor a proof.